We have performed Cu-63 nuclear-magnetic-resonance/nuclear-quadrupole-resonance measurements to investigate the magnetic and superconducting (SC) properties on a "superconductivity dominant" (S-type) single crystal of CeCu2Si2. Although the development of antiferromagnetic (AFM) fluctuations down to 1 K indicated that the AFM criticality was close, Korringa behavior was observed below 0.8 K, and no magnetic anomaly was observed above Tc similar to 0.6 K. These behaviors were expected in S-type CeCu2Si2. The temperature dependence of the nuclear spin-lattice relaxation rate 1/T-1 at zero field was almost identical to that in the previous polycrystalline samples down to 130 mK, but the temperature dependence deviated downward below 120 mK. In fact, 1/T-1 in the SC state could be fitted with the two-gap s(+/-)-wave model rather than the two-gap s(++)-wave model down to 90 mK. Under magnetic fields, the spin susceptibility in both directions clearly decreased below Tc, which is indicative of the formation of spin-singlet pairing. The residual part of the spin susceptibility was understood by the field-induced residual density of states evaluated from 1/T1T, which was ascribed to the effect of the vortex cores. No magnetic anomaly was observed above the upper critical field H-c2, but the development of AFM fluctuations was observed, indicating that superconductivity was realized in strong AFM fluctuations.